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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">f</font> = (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>);<br/>
<a NAME="E1:169"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T118" target="_self" title="FUNCT_7:th.118">Th120</a></span>;<br/>
<a NAME="E2:169"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T121" target="_self" title="FUNCT_7:th.121">Th123</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:169"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:169"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="funct_7.html#T30" target="_self" title="FUNCT_7:th.30">Th32</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D11" title="FUNCT_1:def.11">FUNCT_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> or <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> or <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:169"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:169"><span class="lab"><font color="Green" title="E83">A2</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="169_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:169_1"/>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="169_1_1"><span class="kw">suppose </span></a><a NAME="E1:169_1_1"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c1">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T122" target="_self" title="FUNCT_7:th.122">Th124</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:169_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="funct_7.html#T119" target="_self" title="FUNCT_7:th.119">Th121</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:169_1_1"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="funct_7.html#T32" target="_self" title="FUNCT_7:th.32">Th34</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="169_1_2"><span class="kw">suppose </span></a><a NAME="E1:169_1_2"/><span class="lab"><font color="Green" title="E84">A4</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c2">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T123" target="_self" title="FUNCT_7:th.123">Th125</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:169"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E1:169_1_2"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="ref" href="funct_7.html#T31" target="_self" title="FUNCT_7:th.31">Th33</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="169_1_3"><span class="kw">suppose </span></a><a NAME="E1:169_1_3"/><span class="lab"><font color="Green" title="E84">A5</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c3">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T124" target="_self" title="FUNCT_7:th.124">Th126</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:169_1_3"><span class="lab"><font color="Green" title="E84">A5</font></span></a>, <a class="ref" href="funct_7.html#T120" target="_self" title="FUNCT_7:th.120">Th122</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (1,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:169_1_3"><span class="lab"><font color="Green" title="E84">A5</font></span></a>, <a class="ref" href="funct_7.html#T32" target="_self" title="FUNCT_7:th.32">Th34</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
